Note that the left hand side of (REF ) is independent of the choice of the unitary frame, hence is a globally defined \((1,1)\) -form on \(M^n\) . Fix any point \(p\)  in \(M\) . We may choose a local unitary frame \(e\)  in a neighborhood of \(p\)  so that the matrix \(\theta \)  of the Chern connection \(\nabla \)  under \(e\)  vanishes at \(p\)  (see for example [1]}. Use this frame, then at the point \(p\)  we have \(\overline{\partial }\varphi =0\) , hence
\( \overline{\partial }\eta = \overline{\partial }\eta _i \wedge \varphi _i = \overline{e}_j (\eta _i) \,\overline{\varphi }_j \wedge \varphi _i = \eta _{i,\,\overline{j}}\, \overline{\varphi }_j \wedge \varphi _i.\) 
This proves (REF ). Now let \(k=\ell \)  in Lemma 2 and sum up, we get (REF ).
